Suppose we've arduously calculated the center of mass of a system of 6 masses . And now we want to include another 3 . Do we have to redo the entire calculation? No! A little mathematical manipulation should convince you that we can the calculations of the center of masses of masses 1-6 and 7-9. to calculate the center of mass of masses 1-9 . This is illustrated pictorially:
Call the center of mass of 1-6 . It has a total mass of . Similarly call the center of mass of 1-3 . It has a total mass of . Then what happens if we calculate the center of mass of these two ``pseudo" particles? We get
By pluggin in the definitions of center of mass, you should be able see that you get quite a few cancellations and finally convince yourself that this is equal to
which is just the center of mass of the complete 9 mass system.